Continuity of Linear Operator Between Hilbert Spaces
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Note: Please do not give a solution; I am curious to understand why my solution is incorrect, and would prefer guidance to help me complete the question myself. Thank you. Let $mathcal{H}$ be a Hilbert space, and suppose that $Tintext{Hom}(mathcal{H},mathcal{H})$ . Suppose that there exists an operator $tilde{T}:mathcal{H}rightarrowmathcal{H}$ such that, begin{align} langle Tx,yrangle =langle x,tilde{T}yrangle, end{align} $forall x,yinmathcal{H}$ . Show that $T$ is continuous. My current solution is as follows: Assume for all $delta>0$ there exists $n>Ninmathbb{N}$ such that, begin{align} |x_{n}-x|<delta. end{align} Then, begin{align} langle Tx_{n}-Tx,Tx_{n}-Txrangle &= |Tx_{n}-Tx|^{2}\ &leq|Tx_{n}-Tx|=|T(x_{n}-x)|\ &leq|T||x_{n}-x|rightarrow 0text{